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Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP (O is the origin) and the positive side of the X-axis. Suppose f is a function of the two variables x and y and U cos 0 î + sin 0 ĵ. The directional derivative of f in the direction of U, denoted by D7f, is given by f(x + h cos 0, y + h sin 8) – f(x, y) Dzf(1, y) = lim h→0 h if this limit exists. Note that the directional derivative is a scalar (a real number). The partial derivatives fr and fy are special cases of the directional derivative. Remark (a) Ij U = î, then cos 0 = 1 and sin 0 = 0. Thus, f(r +h u) - flr u)

Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP (O is the origin) and the positive side of the X-axis. Suppose f is a function of the two variables x and y and U cos 0 î + sin 0 ĵ. The directional derivative of f in the direction of U, denoted by D7f, is given by f(x + h cos 0, y + h sin 8) – f(x, y) Dzf(1, y) = lim h→0 h if this limit exists. Note that the directional derivative is a scalar (a real number). The partial derivatives fr and fy are special cases of the directional derivative. Remark (a) Ij U = î, then cos 0 = 1 and sin 0 = 0. Thus, f(r +h u) - flr u)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP (O is the origin) and the positive side of the x-axis. Suppose f is a function of the two variables x and y and U cos 0 î + sin 0 ĵ. The directional derivative of ƒ in the direction of U, denoted by D7#f, is given by f(x + h cos 0, y + h sin 0) – f(x, y) DGf(r, y) = lim h→0 h if this limit exists. Note that the directional derivative is a scalar (a real number). The partial derivatives fr and fy are special cases of the directional derivative. 1 Remark (a) If U = î, then cos 0 = 1 and sin 0 = 0. Thus, f(r + h, y) – f(x,y) D;f(x, y) = lim h-0 h which is the partial derivative of f with respect to x. (6) If U = î, then cos 0 = 0 and sin 0 = 1. Thus, f(x, y + h) – f(r, y) D;f(x, y) = lim h→0 h which is the partial derivative of f with respect to y.
Transcribed Image Text:Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP (O is the origin) and the positive side of the x-axis. Suppose f is a function of the two variables x and y and U cos 0 î + sin 0 ĵ. The directional derivative of ƒ in the direction of U, denoted by D7#f, is given by f(x + h cos 0, y + h sin 0) – f(x, y) DGf(r, y) = lim h→0 h if this limit exists. Note that the directional derivative is a scalar (a real number). The partial derivatives fr and fy are special cases of the directional derivative. 1 Remark (a) If U = î, then cos 0 = 1 and sin 0 = 0. Thus, f(r + h, y) – f(x,y) D;f(x, y) = lim h-0 h which is the partial derivative of f with respect to x. (6) If U = î, then cos 0 = 0 and sin 0 = 1. Thus, f(x, y + h) – f(r, y) D;f(x, y) = lim h→0 h which is the partial derivative of f with respect to y.
Consider f(x, y) = 3r² – 4y² and U = cos n î + sin r î. = COS Find the directional derivative of f in the direction of the U.
Transcribed Image Text:Consider f(x, y) = 3r² – 4y² and U = cos n î + sin r î. = COS Find the directional derivative of f in the direction of the U.
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